Lesson video

Welcome to today's lesson.

My name's Ms. Davies.

I just want to start by saying, well done for choosing to learn using this video.

I hope you find lots of interesting bits as we explore this lesson.

Let's get started.

Welcome to today's lesson on positive rate of change from a graph.

Our learning outcome today.

By the end of the lesson, you'll be able to calculate the positive rate of change, and we're gonna introduce this new word gradient from a graph.

Let's look at some keywords there.

So you might have come across this concept of rate of change already.

So the rate of change is how one variable changes with respect to another.

So for example, if you've been plotting some graphs, you'll have had some X coordinates and Y coordinates, and you might have looked at the relationship between the X and the Y coordinates.

So the rate of change is how the Y coordinates, for example, might change with respect to the X coordinates.

You might have also looked at that if that change is constant, there is a linear relationship.

It's possible that you've come across linear graphs or linear sequences in some of the work that you've done.

And this idea of a constant rate of change is gonna be important today.

Our new word then is the gradient.

We're gonna talk about lots about what the gradient is today, but general, it's a measure of how steep a line is.

So we're gonna look at defining gradient and what gradient is, and then we're gonna look at how we can calculate that from a graph.

So some pupils are drawing staircase patterns across a piece of squared paper.

Here are the beginnings of their patterns.

We've got Andeeps, Izzys, Junes and Lauras.

We'd like you to pause the video and have a think about these questions.

Whose pattern do you think will reach the top first? Whose pattern do you think will reach the top of the page second? And then any assumptions you've made by answering that question.

Off you go.

Let's look at what we've said.

So June's pattern is gonna reach the top first.

You might've said something like, his staircase goes right one square and up two squares.

You might've used this word steepest and said that his staircase is the steepest.

In terms of second well done, if you notice that Andeep and Lauras are moving up at the same rate.

So Andeep and Lauras will reach the top of the page at the same time.

His goes, right one up, one hers goes right two up two.

So they've got that same rate that those staircases are climbing.

And some assumptions you might have thought about, well then if you've got even more.

But you might have thought about the fact that this only works if the piece of papers are all the same height.

The staircases do have to carry on in the same pattern.

Okay, if the pattern of the staircases change, then we won't know who's gonna reach the top first.

So we can describe the steepness of these staircases by looking at the number of squares moved in a horizontal and a vertical direction.

You might have said things like, Izzy staircases move two squares right and one square up.

We're talking about a movement in two directions, horizontal and vertical.

Okay, Aisha has plotted these linear relationships.

What is the same and what is different about her four graphs? What do you think? So here are some of the similarities I came up with.

All her graphs are straight lines.

They all pass through the origin or that zero zero coordinate.

As the X values increase, so do the Y values, or you might have said the graphs go from bottom left to top right.

The differences I came up with.

They have different steepness or different rates of change if you want to use that language.

They rise at a different rates.

So there's just this idea that some of them are steeper than others.

So we can describe how these different graphs rise by looking at how the distance in the Y direction increases compared to the X direction.

So just like with the staircases, we can look at a horizontal and a vertical movement.

If we pick the pink graph, we could say for every two squares in the positive X direction.

So two squares right.

The graph rises one square in the positive Y direction or one square up.

So we go two right one up.

By doing that, we've described a rate of change in the Y direction compared to the X direction.

So Aisha's classmates are now gonna try and describe the rate of change for this graph.

So Andeep says, the graph goes one across, two up.

Izzy says the graph moves one in the X direction and two in the Y direction.

And June says, for every one square in the positive X direction, the graph rises two in the positive Y direction.

Just take a second to think about their descriptions.

Which one do you think is best and can you explain why? I think that June's got the best explanation in this case? Couple of reasons.

He's made it really clear which direction he is referring to.

He says the positive X direction, okay.

If you just say across or in the X direction, we don't know if you're moving to the right or to the left.

So if he's said positive X direction, we've got this really clear idea which way we're going.

He's also described it as a rate of change.

He's used this word for every.

So for every one square in the positive X direction, the graph rises two in the positive Y direction.

That's a rate of change.

We're comparing how far the Y direction moves compared to the X direction.

We're now gonna look at the purple graph and they're all gonna have a go at using this idea of a rate of change.

So this time Andeep says that every three squares in the positive X direction, the graph rises three in the positive Y direction.

Izzy says for every one square in the positive X direction, the graph rises one in the positive Y direction.

And June, for every 10 squares in the positive X direction, the graph rises 10 in the positive Y direction.

You're gonna need a second just to have a look at those three and compare them to the graph.

Who do you think is correct? Right, so this is what Andeep's saying, three right and then three up.

That seems to work fine.

Izzy said one in the positive X direction and one in the positive Y direction, and June said 10 to the right and then 10 up.

Hopefully you saw that all three of them are correct.

They've all described the same overall movement.

They've just used a different number of squares.

So what we're gonna say now then, is it actually useful to refer to the rate of change in the same way? If everyone's talking about a different amount of squares, then it's difficult for us to compare different rates of change.

So what we do is when we talk about rates of change in a graph, we talk about the amount and the Y changes when X increases just by one.

And that measure of rate of change is the gradient.

So what Izzy said, for every one square, right, the graph moves one square up.

That is describing a gradient because it's the rate of change for one movement in the positive X direction.

So let's look at a couple of examples.

So this one here, we would say the gradient of the line is three, because for every one square in the positive X direction, the graph increases by three.

This one here, we'd say the gradient of this line is one 'cause for every one square moved in the positive X direction, the graph increases by one in the Y direction.

So which of these describes the gradient of the line? Read through all four and come up with your own answer.

Good spot if you notice it's the second one.

So increases by two in the Y direction, and here's the key point for every one increase in the X direction.

So always looking at a step of one in the positive X direction when we're talking about gradient.

We're gonna focus specifically today on positive gradient.

So graphs have a positive gradient.

If as X increases, Y also increases.

Let's look at what that looks like.

So all these graphs have a positive gradient because Y is increasing as the graph moves to the right.

They go from kind of the bottom right, bottom left to the top right.

So they're kind of increasing as you move from the left to the right.

So those are all positive gradients.

To counter that these ones do not have a positive gradient.

We're not gonna focus on them today.

But it's useful to be aware that not all graphs have positive gradients.

So quick check.

Which of those four lines A, B, C, or D have a positive gradient? Well, if you spotted that it was A, that pink graph B, the purple graph, and D, the green graph.

C doesn't have a positive gradient because it's decreasing or Y is decreasing as the graph moves to the right.

So we can sometimes see the gradient represented a table of values.

So if you are used to plotting graphs from a table of values, you'll have seen lots of these before.

Notice that our X values have to increase by one if we're gonna spot the gradient, 'cause remember the gradient is for every one increase in the positive X direction.

So if our X values increase by one, then we can see the gradient this time is going to be two.

Because as X increases by one, y increases by two.

If you have a look at this one.

We've got seven, eight, nine, 10, 11 in our X values, that's fine.

They're all increasing by one.

And then our Y values this time are all increasing by four.

So this time the gradients gonna be four as X increases by one, Y increases by four.

So Aisha's looking at this table of values and she reckons the graph of these points has different gradients at different points.

She has made a mistake somewhere.

Can you spot where she has gone wrong? So where she's gone wrong here is that the gradient describes the change in Y for every one increase in X.

If you actually have a look at how our X values increase, they don't have a step of one, which when you're plotting a table of values is absolutely fine if you're then going to use that table of values to plot a graph.

However, if we're looking to spot the gradient, we need to know how they're increasing for every one increase in X.

So true or false, the gradient of the line made by plotting these points is two? Look at the table values in the top right hand corner and make your decision.

Well then if you spotted that one is false.

It looks like the Y values are increasing by two.

But notice the X values do not increase by one.

We can only see the gradient from a table if there are X values are increasing by one.

Time to have a practise.

So starting with the steepest, put these steps in order.

So you're working from the steepest to the least steep from those staircases.

Then draw staircase H, so that is now the steepest.

And letter C another set of steps, join point P to point Q, which have been plotted on some square paper.

Where would those steps go in order of steepness? So you can draw on the page if you wish, and then put them in your order of steepness.

Well done on question one.

So question two, you've got the lines A, B, C, D, and E, and I'd like you to write down which ones have a positive gradient.

Then question three, you've got three different tables of values.

Which tables of values show a relationship which will have a positive gradient when plotted? Off you go and then we'll have a look at our answers.

Well done.

So starting with the steepest, you should have E, then B, then A, then F, then C, then G, and then D is the least steep outta those original staircases.

Lots of different things you could draw for H.

I think by far the easiest is to go one right and then four up.

And then for C, another set of steps join the point P and Q.

They're actually gonna be the same as A.

So however you draw the steps between P and Q ends up being identical steepness to A.

Question two, you've got A, B, and E had the positive gradients.

And then three, A was positive 'cause you can see that Y is increasing as X increases.

B is not positive.

The Y values are going down by four each time.

And then C was positive again.

So as the X values increase, the Y values are increasing as well.

We're now gonna look at calculating positive gradients from graphs.

Let's see how it's done.

So we can calculate the gradient from any graph.

What we need to do is pick points on the line, and draw an arrow to show a step of one in the positive X direction.

Then we can draw an arrow vertically until we reach the line again.

This gradient has a change in X of one and a change in Y of two.

We've gone one right, that's how X has increased and we've gone two up and that gets us back to our line again.

This means that the graph has a gradient of two 'cause Y increases by two when X increases by one.

What do you think the gradient of this line is? You might wanna pause the video and give it a go before we look at the answer.

Okay, so again, we need to pick a coordinate that's on the line.

We're gonna move one in the positive X direction and then count how much we've moved in the positive Y direction.

Hopefully now that I've drawn the arrows on, you can see that this will have a gradient of three.

Y increases by three when X increases by one.

You might be wondering why I've been putting it into a table.

When we look at some slightly trickier gradients, the idea of having a table linking the change X and the change in Y can help us calculate some of those.

So Laura's gonna give this go.

Laura says the gradient of this line is seven.

Let's look at her working out.

Pause the video and have a real close look at what Laura has done.

Is she correct? Well done if you spotted that no, she's actually moved more than one in the positive X direction.

There's more than one square with that arrow going right.

So what advice would you give Laura, so she calculates the gradient correctly? See if you can put it into words.

You might have said something like this.

She should pick a coordinate that lies on an integer coordinate point or at least a point where two lines in her book or on her graph meet.

And that makes it easier to count across one square.

She's definitely gone one and a bit squares.

We don't know exactly how much though.

Laura's gonna give it another go.

So this time she's picked that coordinate.

That's looking better now.

She's moved one in the positive X direction and then five in the positive Y direction.

Her working out is now correct.

What would we say the grading of this line is? Clearly we can see that the gradient is gonna be five.

Right, quick check.

Have a go at matching the graphs to the correct gradients.

Well done.

So that left hand graph has a gradient of three, the middle graph has a gradient of one, and the right hand graph has a gradient of two.

Fantastic guys, you've already shown that you are able to calculate the gradient from a graph.

Okay, so we've got the basics of calculating a positive gradient from a graph.

We're now gonna have a look at a couple of extra bits.

One thing we need to be aware of is the scales on our axis.

It's really important that you read the scales carefully and we label an increase of one in the positive X direction.

Let's have a look at the scales on this axis.

You might be able to see on the x axis, each square is actually 0.

5.

When we calculate the gradient then, we pick our integer point and then we need to make sure that we move one in the positive X direction, not just one square, an actual movement of one along the axis.

Then we need to draw a vertical line back up again.

And just check on your Y axis how much we've actually moved between the numbers.

It's gone from two to four, so that's gonna be a change of two.

So the change in X is one, change in why is two.

If you just looked at that graph straight off, you might think it has a gradient of one.

You might think it's going right one up, one, right, one up one.

But we know now by looking at the scales, actually looking at the numbers on the graph that it goes, right one and up two.

Our gradient then is two.

What do you notice about the scale on these axis this time? Pause and have a good look.

Well dunno if you notice that the x axis this time is going up by one, but the Y axis has a step of five.

It's going up in fives.

We can calculate the gradient just as before.

We just gotta be careful to properly read our axis.

To pick any integer point, I've gone with three zero.

Go across one and this time one square does represent one, and then go up.

It looks like it's gonna be one, but let's have a look at our axis.

We've gone from zero to five.

So that's an increase of five.

So our gradient this time is five.

Quite often we will scale our axis, because if we have really steep graphs, it's not always possible to plot them on a piece of paper.

So often the scales are changed so that we can plot graphs, particularly really steep ones on a physical axis.

So June has tried to calculate the grading of this line, identify his mistakes and what advice would you give him? Brilliant.

I wonder what you said.

So he's got the idea of moving one in the positive X direction, which is really good, but what he needs to do is he needs to actually look at the scale because one square this time is not one.

The question's a little bit tricky bit.

So actually what he needs to do is he needs to move all the way to the one, if you can see it on the far right of the X axis to be a step of one in the X direction.

He also needs to look at the Y axis, because they've been even meaner with our axis this time.

And not only is it five squares that represents one in the X direction, the Y direction is going up in twos.

So we need to look at the number from two to 12 and see that it's actually an increase in 10.

So for every one in the X direction, the graph is rising 10 in the Y direction.

Perfect June's got that right now.

The gradient is actually 10.

So that was a mean question.

Just be really careful when you're working out gradients just to check your axis.

So Izzy wants to calculate the gradient of this line, and the squares are actually quite small to draw her lines on.

So instead of going one right and then going up, she's decided to draw this triangle.

So she's gone right three and six up.

She can still calculate the gradient that's not a problem.

If we put it back in our table that I said was gonna be quite helpful earlier, we've got a change in X, which is three a change in Y, which is six.

What we can do is we can convert that to a gradient.

If for every three in the X direction it goes up six in the Y direction, then we can work out how far it increases in the Y direction for every one in the X direction.

What we can do is we can divide that value by three to get one in our X direction.

We do the same to the other side.

We can say the change in why is gonna be two.

So for every one right is going two up.

You might wanna have a look at your graph and check that that is correct.

What we've used here is called a ratio table.

You might have seen them in other areas of maths.

They work in both directions.

So what we did is we looked at dividing by three to get down from three to one, but also you can actually have a look at the relationship going horizontally.

So what is the relationship to get from three to six? Well that's doubling, isn't it? So we're gonna have to double one to get two.

So you can either work down the ratio table or you can look horizontally.

This is quite a useful method when our gradient is a non-integer value.

So if I draw our one in the positive X direction.

If we now look at our graph, we could have a guess at what the gradient is, but we can't know for sure.

Because we haven't got the minor grid lines, we can't tell exactly whether that's like a half or a third or a quarter.

So what we can do is that we can use this idea of moving further in the X direction and use our ratio table to get that movement down to a gradient.

So here if we look, if we go two in the positive X direction, then it's clearly one in the positive Y direction and that brings us to another your point.

Putting that into a ratio table then, we need our change in X to be one for a gradient.

So if you divide both sides by two, then for every one in the positive X direction, it's going up a half in the Y direction.

Now you might have been able to guess that just from looking at the graph, but you may come across some even trickier fractional gradients where this method is gonna be quite useful.

Gradients can be written as fractions or decimals.

Generally fractions are easier to work with.

So you'll see that I will leave all my gradients as fractions.

So here's an example like I talked about that might not be quite as easy to see.

So again, if I move one in the positive X direction, I can't really see what that movement is in the Y direction.

I know it's less than one, but I'm not sure exactly.

However, if I go three right, then two up will get me back to the line.

If I put that in our ratio table, this time I'm gonna need to divide two by three, which is the fraction two thirds.

Like I said before, a lot easier to leave that as a fraction rather than try and turn it into a recurring decimal.

So what I'd like you to have a go at doing now is fill in the gradient of each line.

I've drawn some ratio tables to help you.

All you to do is work out the change in Y for one positive increase in X.

Off you go.

Perfect.

You should have one.

So the gradient is one.

Moving across the page, you should have a gradient of three.

The change in Y is three.

Bottom left, you should have three over two, and bottom right five over three.

Well done if you listen to what I said and left those as fractions.

Time for a practise then.

So for each one you need to find the grading of each line.

A couple of things that might help you.

If you've got this on paper, you can actually draw those triangles, those arrows onto the graph, and then you can use a ratio table, there's space underneath to draw a ratio table if you wish.

Off you go.

Right second set, same idea.

Feel free to draw a ratio table to help you.

Well done.

Last set.

I do advise you using that ratio table this time as we've got some trickier gradients.

Off you go.

Well done.

You are now definitely an expert in finding the positive gradient from a graph.

Let's have a look at what we got.

So the first one, a gradient of three.

Second one, you should have a gradient of two.

Third one, one, and the fourth one two again.

I decided I didn't need a ratio table for these ones because each graph had a step of one in the X direction.

For E and F so you should have six.

Notice that the graph is going up in twos.

So one right, and then two, four, six up.

Lots of different ways that you could have done F.

You might have noticed that a change in 0.

5 is an increase of one in the Y direction.

And then that means that for every one in the X direction you get a change in Y of two.

Or what you might have done is drawn an arrow that goes two across and two up, and noticed, two squares across that is and two up.

And noticed that that's an increase of one in the X direction and two in the Y direction.

Fantastic thinking.

For G, you get seven.

Again, I've looked at the x axis and noticed that where the one is.

So I've gone across one and then I've counted up and that gives me seven.

And H, this one's a nice one.

It's got a step of one in the X direction, that's fine.

Then you just have to look carefully at your y axis and that's five, 10, 15.

And then the last set.

So the first one you should have a half is two rights and one up.

J you should have a third.

It is three rights and one up.

K, so we need to have a look at our scales this time.

So each square is five.

So we've gone five rights and five, 10, 15 up.

So five rights and 15 up gives us a gradient of three.

And then L again, be very careful that you pick an integer point this time.

So I've picked zero negative three and noticed that if I go three right, then I need to go four up to get to the coordinate three one.

And that gives me a gradient of four over three.

That last one was tricky.

So super impressed if you managed to draw your triangle on accurately and work out a gradient.

Well done guys.

Lots of good thinking there today.

It sets you up really nicely to any other work that you're going to be doing with gradients in the future.

The things that we've come across today.

So gradient measures the amount Y changes when X increases by one.

We've talked lots about positive gradient.

If Y increases, X increases the gradient is positive.

And then we've looked at all sorts of graphs and looking at making sure that we are reading the scales and then we're looking for an increase of one in the positive X direction.

Well done and I look forward to seeing you in our lessons in the future.